Question

计算∫∫z√1-x²-y²

设Ω是由z=√(x^2+y^2)及z=√(1-x^2-y^2)做组成,计算∫∫∫(x+z)dv
答案是π/8,我要过程!

Answer 1

原式=∫dθ∫rdr∫(rcosθ+z)dz (应用柱面坐标变换)
=∫dθ∫rdr*[(rcosθ*+z²/2)│]
=∫dθ∫r[(r√(1-r²)-r²)cosθ+1/2-r²]dr
=∫cosθdθ∫[(r²√(1-r²)-r³)]dr+∫dθ∫(r/2-r³)dr
=[(sinθ)│]*∫[(r²√(1-r²)-r³)]dr+[(θ)│]*[(r²/4-r^4/4)│]
=(0-0)*∫[(r²√(1-r²)-r³)]dr+(2π-0)*(1/8-1/16-0+0)
=0+2π(1/8-1/16)
=π/8